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Transcript
Journal of Oceanography, Vol. 56, pp. 261 to 273, 2000
Tidal Vorticity Around a Coastal Promontory
MOON -JIN PARK 1* and D ONG-PING WANG2
1
2
Department of Oceanography, Chungnam National University, Taejon, Korea
Marine Sciences Research Center, State University of New York, Stony Brook, U.S.A.
(Received 28 January 1999; in revised form 28 September 1999; accepted 28 September 1999)
The vorticity generation around a coastal promontory is examined using a transport
vorticity equation. The complete vorticity balance analysis is made for the transient
and residual vorticity generations around an idealized, symmetric promontory. The
topographic vorticity tendency is found to be the dominant forcing term in the production of transient tidal vorticities. This result is different from the previous works
which emphasized the effects of lateral and bottom frictions. The residual vorticity
balance is between advection and topographic vorticity tendency. The model results
are consistent with the observations off Gay Head, Massachusetts. Also, the stability
of a promontory as well as the offshore sand bank formation are studied by examining the sand transport pattern around the promontory. Strong deposition occurs off
the tip of the promontory while erosion occurs along the upstream slope of the promontory suggesting that an originally symmetric promontory tends to incline cyclonically from the axis normal to the tidal stream. Such tendency is indeed found among
coastal promontories.
Keywords:
⋅ Vorticity generation,
⋅ vorticity balance,
⋅ stability of coastal
promontory,
⋅ topographic
vorticity tendency,
⋅ symmetric promontory,
⋅ asymmetric
residual eddy pair,
⋅ cyclonic inclination,
⋅ sediment transport.
Yanagi (1976, 1978), Oonishi (1977), Pingree (1978),
Pingree and Maddock (1979), Ridderinkhof (1989),
Ridderinkhof and Zimmerman (1990), Signell and Geyer
(1991) used vertically-averaged two-dimensional models to study formation of the tidal residual eddies around
a headland. Tee (1976), Yanagi (1976, 1978) and Oonishi
(1977) suggested that the residual eddy results from
advection of the vorticities generated in a coastal boundary layer. Pingree and Maddock (1979), Ridderinkhof
(1989), Ridderinkhof and Zimmerman (1990) on the other
hand, indicated that the frictional torque is responsible
for the vorticity generation.
Geyer and Signell (1990, 1991) used a ship-board
acoustic Doppler current profiler to map the structures of
transient and residual tidal eddies around Gay Head, a
coastal promontory located at the seaward end of Vineyard Sound, Massachusetts. The Vinyard Sound is about
20 m deep, 15 km wide, and the dimensions of Gay Head
is about 5 km alongshore and 4 km offshore. Signell and
Geyer (1991) simulated the observed transient eddies and
suggested that the transient tidal eddies are formed when
the tidal flow separates from the coast at the headland
tip.
Signell and Geyer (1991), however, did not examine
the vorticity balance, which is necessary to fully understand the generation of tidal eddies. Previously, work on
the vorticity balance has focused on the vorticity associated with the depth-averaged velocity. Park and Wang
1. Introduction
Tidal currents play an important role in dispersing
and transporting materials in shallow coastal seas. Strong
oscillatory tidal currents produce residual currents through
nonlinear interaction with the bottom topography or coastline. For example, a clockwise residual circulation is often found around a sand bank (Huthnance, 1973). Sources
of the tidal residual circulation can be examined by
analyzing the vorticity balance. Tee (1976), Yanagi (1976)
and Pingree (1978) suggested that the nonlinear advection
is the major source of the residual vorticity. Zimmerman
(1978, 1980, 1981) and Robinson (1981, 1983) found that
the tidal residual current is best developed when the topographic length scale is comparable to the tidal excursion
length. They also indicated that the column stretching and
frictional torque are the major sources for instantaneous
tidal vorticities. Effects of the residual current on material dispersion were studied by Zimmerman (1976, 1978,
1986), Uncles (1982) and Loder et al. (1982). They suggested that the residual current combined with the oscillatory tidal currents can give rise to a large dispersion
coefficient.
Strong tidal flows also can produce residual eddies
around a headland or coastal promontory. Tee (1976),
* Corresponding author. E-mail: [email protected]
Copyright © The Oceanographic Society of Japan.
261
(1991, 1994), on the other hand, analyzed the vorticity
associated with the total transport. They showed that it is
advantageous to analyze the transport vorticity whose
pattern can be quite different from the vorticity of the
depth-averaged velocity. They suggested that for transient
tidal vorticity generated over an isolated bottom topography the topographic vorticity tendency is dominant.
The tidal residual eddy may have implications on the
sediment transport. For example, Huthnance (1973),
Zimmerman (1981), and Park and Wang (1994) suggested
a correspondence between the sand bank formation and
tidal residual eddies in shallow coastal seas. Pingree
(1978) and Pingree and Maddock (1979) examined the
formation of a sand bank (Shambles Bank off Portland
Bill, England) around a coastal promontory. They suggested that the sand bank is caused by the tidal vorticities
generated around the promontory and the sand bank tends
to grow faster under a cyclonic vorticity than under an
anticyclonic vorticity. Huthnance (1982a, 1982b) indicated that a sand bank builds up most rapidly when it is
inclined cyclonically relative to the tidal stream direction. Park and Wang (1991, 1994) obtained similar results showing that a circular hollow or bump is unstable
and tends to become elliptic with its principal axis oriented cyclonically to the tidal stream.
In this paper we use the transport vorticity equation
to study the tidal vorticities generated around a coastal
promontory. We consider an idealized case representing
the conditions at Gay Head, Massachusetts. We compute
the transient and residual tidal vorticities, analyze the
transport vorticity balance, and compare our results with
the observations (Geyer and Signell, 1990, 1991) and previous model results (Signell and Geyer, 1990, 1991). We
also examine the processes responsible for offshore sand
bank formation, and investigate the stability and the preferential orientation of a coastal promontory.
2. Model
Tidal vorticities around a coastal promontory are investigated in an idealized coastal ocean. The promontory
is situated in a shallow coastal sea and has dimensions
comparable to Gay Head, Vinyard Sound, Massachusetts.
The Gay Head is about 5 km alongshore and 4 km offshore and slopes to the Sound in about 2 km from the tip
of the headland. The tidal velocities are dominated by
the M2 tide with a typical magnitude of 60 cm s–1 (Geyer
and Signell, 1990). The model ocean is oriented parallel
to the x-axis and closed by northern and southern boundaries (Fig. 1). It is 30 km wide, 50 km long and 20 m deep,
and opens at both ends. The model promontory is 4 km
wide, 4 km long, and has a shallow, submerged flat surface of 2.2 m depth (Fig. 1). The model geometry includes
essential features of Gay Head, but avoids the land boundary which may introduce artificial (i.e. numerical)
262
M.-J. Park and D.-P. Wang
Fig. 1. Schematic diagram of model geometry for a coastal
promontory. The shaded area is the flat top of the promontory of 2.2 m depth and the outer dashed line is the perimeter of the bottom slope. A, B, and C are the locations where
the transient vorticity balance is examined.
vorticities. The model is based on the vertically-integrated
equations of motion and the continuity equation
(Pritchard, 1971);
∂U ∂uU ∂vU
+
+
− fV
∂y
∂t
∂x
(
U U2 + V2
∂ζ
= − gH
−κ
∂x
H2
)
1/ 2
+ Dx
(1)
+ Dy
(2 )
∂V ∂uV ∂vV
+
+
+ fU
∂t
∂x
∂y
(
V U2 + V2
∂ζ
= − gH
−κ
∂y
H2
)
1/ 2
∂ζ ∂U ∂V
+
+
=0
∂x ∂y
∂t
(3)
where u = U/H, v = V/H, H = h + ζ ,
 ∂ 
∂u  ∂   ∂v ∂u   
Dx = AH   2 H  +  H  +    ,

∂x  ∂y   ∂x ∂y   
 ∂x
 ∂   ∂v ∂u   ∂ 
∂v  
Dy = AH   H  +   +  2 H   ,
∂y  
 ∂x   ∂x ∂y   ∂y 
where U and V are the volume transports in x- (alongchannel) and y- (cross-channel) directions, ζ is the sea
surface elevation, h is the water depth, Dx and D y are the
horizontal diffusions in x- and y-directions, κ is a friction
coefficient (=0.0025), A H is horizontal eddy diffusivity
(=100 m2s–1), f is the Coriolis parameter (9.5 × 10–5 s–1),
and g is the gravity acceleration.
Equations (1), (2), and (3) are solved numerically
using a finite element method (Wang, 1975) with a horizontal resolution of 1 km. At the land boundaries the normal velocity is set to zero and a slip boundary condition
is used. At the open boundaries both sea levels and currents are specified using a frictionally-damped progressive Kelvin wave solution (Park and Wang, 1991, 1994).
The model is driven by a tidal wave propagating from
left to right in the positive x-direction with an amplitude
of 1 m and the maximum current speed of about 60 cm
s–1.
3. Transient Vorticity Dynamics
Cross-differentiating Eqs. (1) and (2), we obtain the
transport vorticity equation;
UU 
∂ζ

− κ ∇ × 2  ⋅ k + [ ∇ × D ] ⋅ k
∂t
H 

IV
V
VI
III
( 4)
where Π (=∂ V/ ∂ x – ∂U/ ∂y) is the transport vorticity, N is
the nonlinear advective term, J is the Jacobian operator,
and k is the vertical unit vector. In Eq. (4), term I is the
local acceleration of transport vorticity, term II is the
vorticity advection, term III is the topographic vorticity
tendency, term IV is the vorticity from sea surface divergence, term V is the bottom friction, and term VI is the
vorticity diffusion. This equation may be compared to the
vorticity of the depth-averaged velocity;
uu 
f dH
D
∂ω


− κ ∇ ×
+ ∇ ⋅ (uω ) =
 ⋅ k + ∇ × H  ⋅ k
H dt
H
∂t


A
B
C
D
E
ω=
Π
1
−
(∇H × U ) ⋅ k
H H2
(5)
where ω (= ∂v/∂ x – ∂ u/∂ y) is the vorticity of the depthaveraged velocity. In Eq. (5) the term A is the local acceleration of the vorticity of the depth-averaged velocity,
term B is the vorticity advection, term C is the water column stretching, term D is the bottom friction and term E
is the vorticity diffusion. For low Rossby number flow
with negligible bottom and lateral friction, the vortex
stretching associated with the planetary vorticity, term C,
will be the dominant vorticity production term. The
(6 )
It consists of the depth-average of the transport vorticity
and the correction term. The latter is comparable to the
former for the flow parallel to the isobaths (Park and
Wang, 1991). However, for the flow crossing the isobaths,
it becomes small and the vorticity of the depth-averaged
velocity may be approximated by the depth-average of
the transport vorticity as discussed in Section 5.
Comparing Eq. (4) to Eq. (5), it is evident that the
terms I, II, V and VI of Eq. (4) correspond to terms A, B,
D and E of Eq. (5). That is, the local acceleration, vorticity
advection, bottom friction and vorticity diffusion have
counterparts in both equations. Under the geostrophic
approximation, the terms III and IV in Eq. (4) become
− gJ ( H , ζ ) + f
∂Π
+ [∇ × N ] ⋅ k
∂t
I
II
= − gJ ( H , ζ ) + f
vorticity of the depth-averaged velocity is related to the
transport vorticity;
∂ζ
dH
≈ f
∂t
dt
which corresponds to term C of Eq. (5). This suggests
that the topographic vorticity tendency may play the role
of the column stretching when the transport crosses the
isobaths producing the vorticity. The topographic vorticity
tendency comes from the pressure gradient which is the
dominant term in the equation for the tidal flow and its
effect is shown explicitly in the transport vorticity equation, whereas in the vorticity equation of the depth-averaged flow it is implicit and appears through the continuity equation. Expansion of the friction term V of Eq. (4)
yields,
UU 

−κ ∇ × 2  ⋅ k
H 

U
U
κ
= −κ 2 Π + 2 [U × ∇ U ] ⋅ k − κ 4 U × ∇ H 2 ⋅ k
H
H
H
V −1
V−2
V−3
[
( )]
(7)
The first term on the right hand side of Eq. (7) is the
vorticity dissipation by bottom friction. The second and
third terms are the frictional torque. Thus, Eq. (4) states
that, following a fluid column the transport vorticity is
produced by the topographic vorticity tendency (III),
transport shear (V-2) and depth gradient (V-3), and is dissipated by the bottom friction (V-1) and horizontal diffusion (VI). The major forcing for the transient transport
vorticity is the topographic vorticity tendency, which is
induced when the isolines of the free surface cross the
Tidal Vorticity Around a Coastal Promontory
263
Fig. 2. The instantaneous transport. (a) to (f) are 1/6 tidal cycle interval starting at the time of the maximum ebb. The maximum
transport in (a) is 16.5 m2s –1.
isobaths. The depth-integrated vorticity equation has been
used for the large scale flows over a sloping bottom and
the bottom pressure torque which corresponds to the topographic vorticity tendency in a homogeneous flow has
been shown to be a dominant term in the transport vorticity
equation (e.g. Holland, 1973; Ezer and Mellor, 1994). This
effect is not isolated in the vorticity equation for the depthaveraged velocity (Park and Wang, 1991) and we will
proceed hereafter with the transport vorticity equation.
The instantaneous transport is shown at 1/6 tidal
cycle interval in Fig. 2. The transport pattern at the time
of the maximum ebb (Fig. 2(a)) is similar to that at the
maximum flood (Fig. 2(d)) except the change of the flow
264
M.-J. Park and D.-P. Wang
direction. The transport increases as flows approaching
the tip of the promontory and decreases after flows passing the promontory. The increase of transport is mainly
associated with the increase of currents around the promontory. For a fluid particle moving on a curved path, the
pressure gradient across the fluid particle path is balanced
by the centrifugal and Coriolis accelerations. The relative importance of the centrifugal acceleration with respect to the Coriolis acceleration is measured by the
Rossby number (R o = S/fR). At the tip of the promonory,
S ~ 90 cm s –1 and R ~ 4 km, Ro ~ 2.3. Thus, the centrifugal acceleration is more important than the Coriolis acceleration. As a fluid particle approaches the promontory,
Fig. 3. Transient transport vorticity. (a) to (f) are in 1/6 tidal cycle interval starting at the time of the maximum ebb current. The
solid lines are for the positive and the chain-dotted lines are for zero and the negative values. The contour interval is 15 ×
10 –4 m s–1.
the centrifugal acceleration is balanced by a sea level
depression. The maximum sea level depression is at the
minimum radius of curvature, i.e. off the tip of the promontory. This sea surface slope induces an acceleration
when fluid particles move toward the promontory, and a
deceleration when fluid particles move away from the
promontory. Hence, the maximum velocity is near the tip
of the promontory.
The instantaneous transport vorticity (Fig. 3) shows
a positive vorticity during the ebb and a negative vorticity
during the flood. The maximum positive vorticity is
slightly stronger than the maximum negative vorticity.
Also, the maximum positive and negative vorticities are
found slightly downstream of the promontory. The topographic vorticity tendency, vorticity advective term, bottom friction and vorticity diffusion are examined at three
locations around the tip of the promontory (see Fig. 1 for
the locations). The bottom friction is further divided into
the dissipation by bottom friction (V-1) and frictional
torque (V-2 and V-3). The transient vorticity balance is
shown over a tidal cycle in Figs. 4 and 5. The transient
vorticity is generated by the topographic vorticity tendency and the frictional torque. The former is large off
the tip of the promontory where the sea surface slope is
Tidal Vorticity Around a Coastal Promontory
265
Fig. 4. Transient transport vorticity balance at stations A (top),
B (middle), and C (bottom) (see Fig. 1 for the locations).
The solid lines are for the vorticity advective term, the chaindotted lines are for the topographic vorticity tendency, the
dashed lines are for the friction, and the dotted lines are for
the vorticity diffusion. Note that the values were multiplied
by 10 8.
normal to the bottom slope (Fig. 4). The topographic
vorticity tendency is mostly positive during the ebb and
negative during the flood. The transient vorticity is also
generated by the frictional torque. The frictional torque
due to the depth gradient is positive during the ebb and
negative during the flood (Fig. 5). The depth gradient
contribution is largest off the tip of the promontory, at
the maximum ebb or flood. The frictional torque generated by the depth gradient, on the other hand, tends to be
cancelled by the frictional torque generated by the transport shear and frictional dissipation (Fig. 5). Consequently, the net effect due to bottom friction is small and
the vorticity balance is mainly between the topographic
266
M.-J. Park and D.-P. Wang
Fig. 5. The transport vorticity generation by the frictional torque
at stations A (top), B (middle), and C (bottom) (see Fig. 1
for the locations). The solid lines are for the friction (sum
of terms V-1, V-2 and V-3 of Eq. (7)), the chain-dotted lines
are for the vorticity dissipation by bottom friction (term V1), the dashed lines are for the vorticity generation by transport shear (term V-2), and the dotted lines are for the
vorticity generation by depth gradient (term V-3). Note that
the values were multiplied by 10 8.
vorticity tendency and the vorticity advective term (Fig.
4). The vorticity advective term becomes large at the
maximum flood and ebb. It is almost always positive on
the left-hand side of the promontory, as there is an influx
of positive vorticity during the ebb and an efflux of negative vorticity during the flood. Similarly, it is mostly negative on the right-hand side.
4. Residual Vorticity Dynamics
The residual transport vorticity and the residual transport are obtained by averaging their respective instanta-
Fig. 6. (a) Residual transport vorticity. The solid lines are for the positive and the chain-dotted lines are for zero and the negative
values. The contour interval is 2.5 × 10–4 m s –1. (b) Residual transport. The maximum value is 2.0 m2s –1.
Fig. 7. Residual transport vorticity balance. The solid lines are for the positive and the chain-dotted lines are for zero and the
negative values. The contour interval is 7 × 10 –8 m s –2. (a) Residual vorticity advective term, (b) residual topographic vorticity
tendency, (c) residual friction, and (d) residual vorticity diffusion.
neous values over a tidal cycle. The residual vorticity is
positive (cyclonic) on the left-hand side of the promontory and negative (anticyclonic) on the right-hand side
(Fig. 6(a)). The residual transport pattern shows a cyclonic
eddy on the left-hand side of the promontory and an anticyclonic eddy on the right-hand side (Fig. 6(b)). Between
the counter-rotating eddy pair, there is a strong offshore
jet off the tip of the promontory. Because of the larger
positive vorticity, the residual transport is stronger in the
cyclonic eddy than in the anticyclonic eddy.
In order to understand how the residual transport
vorticity arises, we analyzed the residual vorticity balance. Averaging Eq. (4) over a tidal cycle, we obtained
the residual transport vorticity equation. This consists of
Tidal Vorticity Around a Coastal Promontory
267
the residual vorticity advective term, residual topographic
vorticity tendency, residual friction, and residual vorticity
diffusion;
[∇ × N ] ⋅ k
+ gJ ( H , ζ )
UU

= − κ ∇ × 2
H


 ⋅ k + [∇ × D ] ⋅ k

(8)
at the right-hand side (Fig. 7(a)). On the other hand, the
residual topographic vorticity tendency is positive at the
right-hand side and negative at the left-hand side (Fig.
7(b)), which tends to counterbalance the residual vorticity
advective term. In other words, these are two dominant
terms in the residual vorticity balance. The residual friction (Fig. 7(c)) and residual horizontal diffusion (Fig.
7(d)) are relatively small.
5.
where 〈 〉 represents the time average over a tidal cycle.
The dominant advective term is 〈u·∇Π〉 which, due to its
nonlinearity, has a large residual when averaged over a
tidal cycle. As noted before, the vorticity advective term
during a tidal cycle is mostly positive at the left-hand side
of the promontory and negative at the right-hand side (Fig.
4). Thus, the residual vorticity advection is positive (cyclonic) at the left-hand side and negative (anticyclonic)
Comparison of Model Results with the Observations
The model results can be compared with the observations at Gay Head (Geyer and Signell, 1990, 1991).
Figure 8 shows the instantaneous vorticity ( ω = ∂v/ ∂x –
∂ u/∂ y) and current at the maximum flood from observations and model results. The calculated vorticity pattern
is quite similar to the observation. Both show a negative
vorticity off the tip of the promontory having the maxi-
Fig. 8. Comparison of the observed instantaneous vorticity of depth-averaged velocity, and the observed currents (Geyer and
Signell, 1991) with those from the model at the time of the maximum flood. The contour interval of the vorticity is 0.5 ×
10 –4 s–1. The maximum velocity from the model is 0.87 m s–1.
268
M.-J. Park and D.-P. Wang
mated by the depth-average of the transport vorticity. The
residual currents from the observations show a cyclonic
eddy at the left-hand side of the promontory and an anticyclonic eddy on the right-hand side (Fig. 9). The offshore extent of the jet is about 5 km, and the maximum
residual current is about 25 cm s–1. The model results reveal a comparable pattern. However, the observations
indicate a symmetric eddy pair, whereas the model shows
an asymmetric pair with stronger cyclonic eddy.
In the model the axis of the promontory is normal to
the tidal stream, and hence the centrifugal force acting
on flood currents is almost equal to that on ebb currents.
Due to the opposing effect of Coriolis force on flood and
ebb currents, however, the cyclonic eddy becomes
stronger. On the other hand, the axis of Gay Head is inclined cyclonically towards the tidal stream axis. This
provides a sharper curvature and a stronger centrifugal
force for the flood current (Geyer, 1993). Perhaps, the
oblique orientation of the headland axis to the tidal flow
leads to a more symmetric residual eddies.
Fig. 9. Comparison of the observed residual currents with the
residual currents from the model. The maximum velocities
are 25 cm s–1 from the observed data (Geyer and Signell,
1991) and 21 cm s –1 from the model.
6. Sediment Dynamics
The tidal vorticities may play an important role in
the sediment transport and sand bank formation. Previous studies on sediment transport around a coastal promontory were rather qualitative (Pingree, 1978; Pingree and
Maddock, 1979), and were confined to the study of offshore sand bank formation (Pingree, 1978; Pingree and
Maddock, 1979; Heathershaw and Hammond, 1980). We
used a more quantitative approach to examine effects of
the tidal vorticities on the sand transport around a coastal
promontory and on the stability of the promontory itself.
Following Huthnance (1982a, 1982b) and Park and Wang
(1991, 1994), we assumed that the bed load transport rate
Qb (volume/unit width/time) is related to |u|3 ,
Qb ~ u (u + λ u ∇h)
2
mum vorticity located at slightly downstream off the
promontory. The model and observed vorticity dimensions
also are comparable, about 8 km alongshore. The observed
vorticity (–4.5 × 10–4 s –1), however, is about two times
the model result (–2.4 × 10–4 s–1). By comparison, Signell
and Geyer (1990, 1991) obtained the maximum vorticity
for the flood current of –25 × 10–4 s–1 from their model.
Also, both observed and modelled currents for the maximum flood show clearly the crossing of the isobaths where
the maximum vorticity is located (Fig. 8). It is worthwhile to note that the pattern of the vorticity of the depthaveraged velocity at the maximum flood is similar to that
of the transport vorticity (Fig. 2(d)). This results from
the fact that the correction term on the right-hand side of
Eq. (6) is smaller than the depth-average of the transport
vorticity for the flow crossing the isobaths. Thus, the
vorticity of the depth-averaged velocity may be approxi-
( 9)
where the second term on the right-hand side of Eq. (9) is
the enhancement of the down-slope transport and λ of
0.005 is used as in Huthnance (1982a, 1982b).
We first examined the transient sediment transport
process by computing the instantaneous divergence of bed
load (∇·Qb ). The erosion is expected in areas of bed load
divergence and the deposition in areas of bed load convergence. During the maximum ebb current, the velocity
increases over the broad area of the slope toward the tip
of the promontory, and the erosion occurs over the slope
along the right-hand side of the promontory (Fig. 10(a)).
As the ebb current passes over the promontory, however,
the velocity decreases abruptly, and the deposition is concentrated behind the tip of the promontory resulting in an
asymmetric pattern of erosion and deposition (Fig. 10(a)).
During the maximum flood current, on the other hand,
Tidal Vorticity Around a Coastal Promontory
269
Fig. 10. Instantaneous bed load divergence. (a) to (f) are in 1/6 tidal cycle interval starting at the time of the maximum ebb
current. The solid lines are for the positive and the chain-dotted lines are for zero and the negative values. The contour
interval is 3 × 10–5 m 2s–3.
the velocity changes across the tip of the promontory are
more gradual. Thus, the erosion and deposition occur over
a broad area, and the sedimentation pattern tends to be
much more symmetric (Fig. 10(d)). The different patterns
of erosion and deposition between the maximum flood
and ebb currents are a manifestation of the asymmetric
tidal currents. The centrifugal acceleration is partly
canceled by the Coriolis acceleration during the flood,
but it is reinforced during the ebb. This creates a stronger
positive vorticty during the ebb currents, which results in
a more concentrated deposition near the tip of the promontory.
270
M.-J. Park and D.-P. Wang
The residual bed load divergence was obtained from
averaging over a tidal cycle the instantaneous values. It
has a net erosion along the left-hand side (upstream) slope
of the promontory and a net deposition on the slope off
the tip of the promontory (Fig. 11(a)). The deposition also
occurs outside the slope on areas where the residual eddies are formed (Fig. 11(b)), but the deposition rate is an
order of magnitude smaller than that off the tip of the
promontory. Previous studies (e.g. Pingree and Maddock,
1979) discussed only the relationship between the residual
eddies and the offshore bank formation around a promontory. The result of this study shows that the net sedi-
Fig. 11. Residual bed load divergence. The solid lines are for
the positive and the chain-dotted lines are for zero and the
negative values. (a) The contour interval is 10 × 10 –6 m2s –3.
(b) The contour interval is 1 × 10–6 m 2s–3.
ment transport associated with the coastal promontory
itself is an order of magnitude larger than that for the
offshore bank formation. This net erosion and deposition
pattern suggests that the promontory will tend to incline
cyclonically from the axis normal to the tidal stream. Indeed, as discussed earlier, Gay Head is such an example.
The cyclonic inclination of a promontory is also found at
Portland Bill in English Channel. Also, for an asymmetric promontory inclining cyclonically from the axis normal to the tidal stream, the centrifugal acceleration is
enhanced for the flood current and reduced for the ebb
current. This may result in more symmetric residual eddies, as found in Portland Bill (Pingree, 1978; Pingree
and Maddock, 1979) and Gay Head (Geyer and Signell,
1990). In other words, an asymmetric promontory may
be a more stable configuration (Park, 1998) with a symmetric eddy pair. On the other hand, a symmetric promontory with the axis orientation normal to the tidal stream
may not be stable until it deforms into a shape by which
the resulting residual eddies become symmetric.
7. Discussion
The transport vorticity equation was used to study
the topographic vorticity generation by the tidal currents
around a coastal promontory in shallow coastal seas. Our
results indicate the presence of strong transient vorticities
off the tip of a promontory. The vorticity analysis shows
that the transient vorticities are generated mainly by the
topographic vorticity tendency, which is consistent with
the results on other isolated bottom topographic features
such as hollow and sand banks (Park and Wang, 1991,
1994). In contrast, previous model studies suggested forcing by lateral friction or frictional torque. Tee (1976), for
example, indicated that a no-slip boundary condition is
essential for the vorticity generation around a promontory. Yanagi (1976, 1978) and Oonishi (1977) also emphasized the lateral friction. Signell and Geyer (1991)
adopted similar approach, but they used higher resolution. Our study, however does not depend on a “wall”
boundary. Since we usually find in nature a bottom sloping upward to the coast in a coastal promontory as in our
study, this mechanism may not be different from the bottom frictional torque (Zimmerman, 1981).
Pingree and Maddock (1979), Ridderinkhof (1989)
and Ridderinkhof and Zimmerman (1990), on the other
hand, suggested that the frictional torque is important for
the transient vorticity generation around a coastal promontory. We found that the frictional torque associated with
the depth gradient is indeed large. However, it is counterbalanced by the frictional torque associated with the
transport shear and the net effect is smaller than the topographic vorticity tendency. In fact, it was also noted that
the vorticity generation by the depth gradient is opposed
by the velocity shear (Pingree and Maddock, 1979), and
the combined effect of the vorticity generation by the
depth gradient and the velocity shear in the vorticity equation of the depth-averaged flow is small (Ridderinkhof,
1989; Ridderinkhof and Zimmerman, 1990). On the other
hand, the magnitude of the column stretching associated
with the relative vorticity (ω ∇·u) was found to be comparable to the combined effect of the frictional torque
(Ridderinkhof, 1989). This column stretching mechanism
may play a role analogous to the topographic vorticity
tendency for the vorticity generation around a coastal
promontory where the advective acceleration is more
important than the Coriolis acceleration (Ro ~ 2.3 in our
study). The residual vorticity balance of the vorticity of
the depth-averaged flow also shows that the vorticity
advection is largely balanced by the column stretching
(Ridderinkhof, 1989) which is analogous to the residual
transport vorticty balance. These results appear to be consistent with the fact that the vorticity of the depth-averaged velocity in this study may be approximated by the
depth-average of the transport vorticity as described in
Section 5.
Tidal Vorticity Around a Coastal Promontory
271
The present study investigates a coastal promontory
that is quite typical in shallow coastal seas. The length
scale of the promontory is comparable to the tidal excursion amplitude, that is, the inertia ratio is O(1). This is an
essential condition for the maximum vorticity generation
(e.g. Zimmerman, 1981; Robinson, 1981). The Reynolds
number (R e = H/ κw) is about 4.2 and the inertia ratio
(K = U/ σw) is about 2.1 in our study, where σ is the tidal
frequency and w is the width scale of the promontory.
Since the Reynolds number is comparable to the inertia
ratio, the frictional decay length (L = H/2κ ~ 4.2 km) is
comparable to the tidal excursion (E = 2U/σ ~ 8.5 km).
In other words, while the time-dependent effects are important, friction is strong enough so that the vorticity decays over the course of a tidal cycle (Signell and Geyer,
1991). Increasing the water depth increases the Reynolds
number, that is, the advective acceleration will dominate
the friction. Signell and Geyer (1991) have explored the
flow dynamics in the high Reynolds number limit. Although the dynamics of flow separation is quite interesting, it is clearly beyond the scope of the present work.
Increasing tidal velocity or promonotory curvature (i.e.
to increase the Rossby number) will make the advective
acceleration much more important than the Coriolis acceleration, and the residual eddies may become more symmetric.
The three-dimensional effect on the sand transport
is ignored in this study. On the other hand, the secondary
circulation with offshore flow near the surface and compensating return flow near the bottom is common around
the tip of a headland (Heathershaw and Hammond, 1980;
Geyer, 1993). Deleersnijder et al. (1992) and Wolanski
et al. (1996) used three-dimensional models to investigate the flow structure around a small island. They suggested that upwelling may occur in the center of eddies.
However, the magnitude of the secondary circulation is
an order of magnitude smaller than the primary current
(Heathershaw and Hammond, 1980; Deleersnijder et al.,
1992; Geyer, 1993). This suggests that the secondary circulation effect may be of minor importance in this study
of sand transport. Nevertheless, to fully explore the threedimensional effect of the flow on the sediment transport
around a coastal promontory, we may need a three-dimensional model.
Acknowledgements
This study was partially supported by the grants
(1998-022-H00009 and 1997-013-147) from Korean Ministry of Education. M.-J. P. would like to appreciate the
hospitality of the personnel of Marine Sciences Research
Center, State University of New York, Stony Brook during his visit as a visiting professor in 1997–1998 academic year where a part of this study was done. We also
appreciate useful comments from reviewers.
272
M.-J. Park and D.-P. Wang
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